\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
Test:
NMSE example 3.6
Bits:
128 bits
Bits error versus x
Time: 8.4 s
Input Error: 19.6
Output Error: 0.7
Log: ⚲
Profile: 🕒
\(\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{{\left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x + 1}}\right)}^2}\)
  1. Started with
    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
    19.6
  2. Using strategy rm
    19.6
  3. Applied frac-sub to get
    \[\color{red}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}} \leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]
    19.6
  4. Applied simplify to get
    \[\frac{\color{red}{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \leadsto \frac{\color{blue}{\sqrt{1 + x} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
    19.6
  5. Using strategy rm
    19.6
  6. Applied flip-- to get
    \[\frac{\color{red}{\sqrt{1 + x} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}} \leadsto \frac{\color{blue}{\frac{{\left(\sqrt{1 + x}\right)}^2 - {\left(\sqrt{x}\right)}^2}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
    19.4
  7. Applied simplify to get
    \[\frac{\frac{\color{red}{{\left(\sqrt{1 + x}\right)}^2 - {\left(\sqrt{x}\right)}^2}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}} \leadsto \frac{\frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
    0.4
  8. Using strategy rm
    0.4
  9. Applied add-sqr-sqrt to get
    \[\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x} \cdot \color{red}{\sqrt{x + 1}}} \leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x} \cdot \color{blue}{{\left(\sqrt{\sqrt{x + 1}}\right)}^2}}\]
    0.4
  10. Applied add-sqr-sqrt to get
    \[\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\color{red}{\sqrt{x}} \cdot {\left(\sqrt{\sqrt{x + 1}}\right)}^2} \leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{{\left(\sqrt{\sqrt{x}}\right)}^2} \cdot {\left(\sqrt{\sqrt{x + 1}}\right)}^2}\]
    0.7
  11. Applied square-unprod to get
    \[\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\color{red}{{\left(\sqrt{\sqrt{x}}\right)}^2 \cdot {\left(\sqrt{\sqrt{x + 1}}\right)}^2}} \leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{{\left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x + 1}}\right)}^2}}\]
    0.7

Original test:


(lambda ((x default))
  #:name "NMSE example 3.6"
  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1))))
  #:target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))